Theorem nfrexya 2574

Description: Not-free for restricted existential quantification where 𝑦 and 𝐴 are distinct. See nfrexw 2572 for a version with 𝑥 and 𝑦 distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)

Hypotheses

Ref	Expression
nfralya.1	⊢ Ⅎ𝑥𝐴
nfralya.2	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfrexya	⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑

Distinct variable group:   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem nfrexya

Step	Hyp	Ref	Expression
1		nftru 1515	. . 3 ⊢ Ⅎ𝑦⊤
2		nfralya.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	a1i 9	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
4		nfralya.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	4	a1i 9	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
6	1, 3, 5	nfrexdya 2569	. 2 ⊢ (⊤ → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑)
7	6	mptru 1407	1 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑

Syntax hints:  ⊤wtru 1399  Ⅎwnf 1509  Ⅎwnfc 2362  ∃wrex 2512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517

This theorem is referenced by:  nfiunya  4003  nffrec  6605  nfsup  7234  caucvgsrlemgt1  8058  zsupcllemstep  10535  nfsum1  11979  bezout  12645